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Jan 31 at 17:04 vote accept Mohammad Ghomi
Jan 31 at 3:39 comment added Mohammad Ghomi Thanks again, I got it now.
Jan 31 at 3:29 comment added Anton Petrunin @MohammadGhomi yes and yes; this example is isometric to a solid quadrangle in a 2-dimensional cone with nonpositive curvature. The latter can approximated by a smooth metric on the plane with curvature supported in a small neighborhood of one point.
Jan 30 at 23:32 comment added Mohammad Ghomi Thanks Anton, are the triangles glued only along the side $x\overline{y}$? Are there such examples which are Cartan-Hadamard manifolds, or at least geodetically complete?
Jan 30 at 21:53 history answered Anton Petrunin CC BY-SA 4.0